I was able to work around the problem by using "regedit" to delete the registry key:

HKEY_CLASSES_ROOT\Applications\Rhino.exe

as both Administrator and normal user. Then, choose a file that you want opened by default with Rhino 5 and go through the usual "Open With" process, but use the "Browse" option to set the path to Rhino 5. Note that you will have to choose exactly one of the 32-bit or 64-bit versions, since both have executables named "Rhino.exe". Once done, Rhino 5 should show up as an icon for future "Open With" dialogs.

The problem was that the paths under this key referred to the WIP (as normal user) and Beta (as administrator) paths, not the release version paths. Because the paths were wrong, the "Open With" dialog could not access an icon for Rhino 5, and therefore omitted it from the list. Moreover, the "Browse" option will not update the paths under that registry key, even if different, so we are stuck deleting the key and starting over.